Reconstruction of 3-D coordinates

 Given that the cameras have been calibrated and the two perspective projection matrices C =[q_ij] and C'=[q'_ij] are known, then for any scene point M with unknown 3-D coordinates (X,Y,Z), that projects onto the two retinal planes at (u,v) and (u',v'), we have

$$\mbox{\bf C} \left[ \begin{array} {c} X \\ Y \\ Z \\ 1 \e... ...t[ \begin{array} {c} s' u' \\ s' v' \\ s' \end{array}\right].$$

Eliminating s and s' and combining the two equations into matrix form gives

$$\left[ \begin{array} {ccc} q_{11}-uq_{31} & q_{12}-uq_{32} ... ...- q_{24} \\ u' - q'_{14} \\ v' - q'_{24} \end{array}\right].$$

This is a linear system in (X,Y,Z). The 3-D coordinates of M can be easily computed.